GardaWorld said when it announced the offer earlier on Monday that it had encouraged G4S shareholders to "mandate their board's engagement" after its attempts to engage directly were "summarily dismissed or ignored on three occasions".
Shares in G4S had jumped 25% to 182.3 pence by 1341 GMT, just shy of the 190 pence per share offer, which represents a premium of about 30% to its last close.
G4S, whose shares had lost around 33% of their value this year to Friday's close, rejected the sweetened proposal made on Sept. 1, revealing that it had also declined two previous offers in June for 145 pence per share and 153 pence per share each.
"The Board believes that the timing of the proposal is highly opportunistic, coming as it does at a time of severe turbulence in global financial markets," it said, urging its shareholders to not take any action.
GardaWorld last year considered a cash offer for some or all of G4S, but by May had opted not pursue a deal. Two months later private equity BC Partners bought a majority stake in the Canadian firm.
In the past year G4S has had to allay investor concerns after setbacks including charges that some executives had defrauded the Ministry of Justice, the loss of a contract to run a Birmingham prison, and Norway's wealth fund shunning investments in the firm.
It sold off most of its cash-handling business in February to U.S. peer Brinks Co, and in July announced plans to lay off some employees at its retained UK cash operations, which have attached pension obligations.
"G4S needs an owner, not a manager," GardaWorld's founder and chief executive Stephan Cretier said on Monday. He said he understood G4S's importance as a UK employer and steps would be taken to address its "underfunding" of UK pension obligations.
"Since the pandemic G4S's valuation has made it more appealing, whilst revenues of about 7 billion pounds annually remain far ahead of GW," said Markets.com analyst Neil Wilson. "This will be the tiddler swallowing the whale."
($1 = 0.7772 pounds)
By Yadarisa Shabong